Applied Arithmetic and Measure

4.6 APPLIED ARITHMETIC AND MEASURE

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 1

TITLE: MEASUREMENT OF LINES AND CURVES
TOPIC: MEASURE
AIM: Students are encouraged to measure various objects accurately using tape or ruler. This helps them to develop a feel for size and an understanding of units, and increases their power of estimation. (How many students can describe their height, even approximately, in centimetres?)

RESOURCES:
Various objects, most of which will be either in the students' school bags or in their classroom. Ruler and tape measures. Copybook to record results.

METHOD:
Students are urged to measure the length and breadth of their textbooks, calculators, school desks, and so forth. Problems such as "How could the thickness of a single page of the textbook be estimated?" could be proposed and solved.
Follow-up work could be along the following lines:

1. With the use of photocopying machines it is possible to provide student work-sheets containing lines, both straight and curved, the lengths of which are to be determined. The measurement of the curved line might be an opportunity for class discussion and dialogue.
2. Outdoor area activities: Area calculations of the following could be made as they are in everyday use: basketball/tennis/volleyball courts, football pitches.
3. Car-park space: determine the maximum number of cars that can be comfortably parked in the school car-park.
4. Students are asked to peg out a rectangular area of 6m x 5m. A rope with knots equally spaced and shaped in the form of a 3, 4, 5 triangle will provide an accurate rightangle.
   The accuracy of this experiment can be tested by measuring the two diagonals.
5. The previous field-work can be related to the pegging out of a site for a new house. If one of the students in the class has a parent or relative involved in house building then this could be an ideal opportunity to invite that person in to show how this work is achieved (most likely now using a theodolite with a rotating telescope). The opportunities for discussion using mathematical language are immense when using the "Guest Speaker" methodology.
6. The diameter (and hence radius) of a small circular object may be found by placing a number of such objects, for example small coins, in a straight line, measuring the total length l and dividing by the number of objects n. This gives greater accuracy than would be obtained by measuring one such object on its own.

CLASSROOM MANAGEMENT IMPLICATIONS:
Many of the follow-up suggestions involve activities outdoors. The obvious benefit is that students can begin to relate mathematics to the real world.

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 2

TITLE: DISCOVERING p
TOPIC: INTRODUCTION TO p
AIM: To give students an appreciation of "pi", how it might have come about and its use in formulae.

RESOURCES:
1 wheel (disc)
1 sheet of cardboard
Some paint
Two books

METHOD:
Place a mark on the wheel.
Roll the wheel on the cardboard and indicate on the cardboard the two positions that correspond to the mark on the wheel (see diagram).
Measure the distance between the two marks.
Find the diameter using two books, as shown below.
Divide the diameter length into the result obtained above.
This should give 3.14 approximately.
Follow-up homework can be along the following lines:
Select a cylindrical object at home, e.g. a can of beans.
Measure the circumference with a piece of string.
Measure the diameter using books or blocks as above.
Calculate the ratio:
(length of circumference)/(length of diameter)
Repeat a number of times with different cylinders.
Calculate the mean value, to give an approximation to p.

The following table can be used for recording purposes:
Complete the table:

CLASSROOM MANAGEMENT IMPLICATIONS:
This activity can be done individually by students or in pairs.

NOTE:
The approximate value of "pi" will be close to 3.14 and students will soon begin to appreciate the concept of approximations and why the value of "pi" is still undetermined.

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 3

TITLE: ARRIVING AT pr2
TOPIC: AREA OF A CIRCLE
AIM: To demonstrate an idealised way of finding the area of a circle.

RESOURCES:
Coloured card, scissors and geometry set.

METHOD:

1. Cut out a coloured disc, radius r, and mark out eight sectors as shown.
2. Cut out the eight sectors and slot them together to form a "bumpy parallelogram" as in the diagram.
3. The students should be challenged to give the approximate lengths of the sides. They should use their previous knowledge of the relationship between the diameter (D) and the circumference (C).
   C = pD
   C = p.2r
   1/2C = pr
4. The outcome of the work should be knowledge of how the formula "area = pr2" came into existence, its meaning and an enhanced facility to remember the formulae.
5. The final diagram should be cut out of coloured card and will make a useful wall-display as an aide-memoire for all classes.

CLASSROOM MANAGEMENT IMPLICATIONS:
This activity can be done individually by students or in pairs.

NOTE:

1. Dexterity and accurate measuring will be essential in this activity and the students should show a pride in presenting their finished drawings.
2. Mathematical terms like sector, circumference, approximation, formulae, division and sub-division should be frequently used so that such words become part of the students' lexicon.

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 4

TITLE: FINDING THE VOLUME OF A SPHERE
TOPIC: VOLUME
AIM:

1. To find the volume of a sphere using a "hands-on" approach.
2. To verify result using the "formula" approach.
3. To use both cm3 and ml as units of volume.

RESOURCES:
Golf ball, two books, ruler, graduated cylinder, elastic bands.

METHOD:
Measure the diameter of the golf ball.
Determine the radius r.
Calculate the volume using the formula Volume = p r3
Check the result as follows.
Approximately half fill a graduated cylinder with water.
Mark the level of water with an elastic band or marker.
Note the volume V1.
Carefully add the golf ball to the graduated cylinder.
Mark the new height. Note V2.
Calculate V2 - V1.
Compare with the original calculation for the golf ball using the formula.
The following table can be used to record the results.

Results

CLASSROOM MANAGEMENT IMPLICATIONS:
This is an excellent opportunity to make a cross-curricular link with science teachers in the school. A visit to the science laboratory for this methodology can be arranged for more comfort.
This activity can be done individually by students or in small groups. Group work offers more opportunities for communication, dialogue and inquiry.

NOTE:
Homework can be used to follow up this work by asking students to repeat the experiment at home using a measuring jug. (Remember, one millilitre or 1 ml is the same as 1 cm3.)

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 5

TITLE: MEASURING THE SURFACE AREA OF A CYLINDER
TOPIC: AREA
AIM: To introduce students to the formula for the area of a cylinder.

RESOURCES:
Any cylindrical object and a sheet of paper.

METHOD:
The curved surface area of cylinders can be measured by wrapping a suitably folded sheet of paper around the cylinder. The area of the folded sheet can then be easily worked out using the familiar formula:
Area = l × w
Students will see readily that the length l = circumference = 2pr and that the width w = h
Thus, area = 2pr × h = 2prh

CLASSROOM MANAGEMENT IMPLICATIONS:
This activity can be done individually by students or in pairs.

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 6

TITLE: GETTING A FEEL FOR VOLUME
TOPIC: VOLUME
AIM: To introduce students to the volume of common objects and associated units.

RESOURCES:
Milk cartons, cubes, cola cans, tennis balls and golf balls, calculator.

METHOD:
Solids occupy space and students should be encouraged to "experience" or "feel" the volume phenomenon and appreciate that it has associated units which differ from those of length or area.

1. Start by taking a rectangular box-shaped milk carton (1 litre capacity) - it provides the student with a clear picture of what a litre is. [Students might like to know that the volume of the average student is equivalent to approximately fifty of these cartons.]
2. It can be pointed out to students that 1 litre = 1000 cm3.
   A small piece of plasticine shaped into a cube of side 1 cm can serve as a visual aid for 1 cm3 - it is literally a one centimetre cube.
3. Repeat this exercise with other common objects as mentioned above.

CLASSROOM MANAGEMENT IMPLICATIONS:
Follow-up work could involve some or all of the following activities:

1. A soft-drinks can approximates to a perfect cylinder. Its volumetric content may be read from the attached label or measured directly using a graduated cylinder. Does calculation produce the same result using the formula pr2h? The calculator will help enormously to take the drudgery out of these calculations and students will also learn the value of rounding off answers.
2. Golf balls or tennis balls are suitable for the study of spheres (for younger classes use a basketball/football/volleyball). The diameter can be measured using a sandwich method and ruler.

NOTE:
Archimedes is credited with showing that the surface area of a sphere is equal to the curved surface area of the smallest cylinder that contains the sphere.

APPLIED ARITHMETIC AND MEASURE LESSON IDEA 7

TITLE: PERIMETER AND AREA
TOPIC: PERIMETER AND AREA OF VARIOUS FIGURES
AIM: To give students practice at finding the area and perimeter of various figures.

RESOURCES:
The grid below, more copies of which can be constructed easily from a standard word processing package that incorporates a basic set of drawing tools.

METHOD:
Examine the four shapes labelled A,B,C,D and complete the table below.
Insert the perimeter and area of each shape in the space provided in the table below.
The perimeter of A is given.

CLASSROOM MANAGEMENT IMPLICATIONS:
None